Dimension of the SLE light cone, the SLE fan, and SLE${}_{\kappa }(\rho )$ for $\kappa \in (0,4)$ and $\rho \in [\frac{\kappa }{2}-4,-2)$

Abstract

Suppose that $h$ is a Gaussian free field (GFF) on a planar domain. Fix $\kappa \in (0,4)$. The SLE${}_{\kappa }$ light cone $\mathbf{L}(\theta )$ of $h$ with opening angle $\theta \in [0,\pi ]$ is the set of points reachable from a given boundary point by angle-varying flow lines of the (formal) vector field $e^{ih/\chi }$, $\chi =\frac{2}{\sqrt{\kappa }}-\frac{\sqrt{\kappa }}{2}$, with angles in $[-\frac{\theta }{2},\frac{\theta }{2}]$. We derive the Hausdorff dimension of $\mathbf{L}(\theta )$. If $\theta =0$ then $\mathbf{L}(\theta )$ is an ordinary SLE${}_{\kappa }$ curve (with $\kappa <4$); if $\theta =\pi$ then $\mathbf{L}(\theta )$ is the range of an SLE${}_{{\kappa }^{\prime }}$ curve (${\kappa }^{\prime }=16/\kappa >4$). In these extremes, this leads to a new proof of the Hausdorff dimension formula for SLE. We also consider SLE${}_{\kappa }(\rho )$ processes, which were originally only defined for $\rho >-2$, but which can also be defined for $\rho \le -2$ using Lévy compensation. The range of an SLE${}_{\kappa }(\rho )$ is qualitatively different when $\rho \le -2$. In particular, these curves are self-intersecting for $\kappa <4$ and double points are dense, while ordinary SLE${}_{\kappa }$ is simple. It was previously shown (Miller-Sheffield, 2016) that certain SLE${}_{\kappa }(\rho )$ curves agree in law with certain light cones. Combining this with other known results, we obtain a general formula for the Hausdorff dimension of SLE${}_{\kappa }(\rho )$ for all values of $\rho$. Finally, we show that the Hausdorff dimension of the so-called SLE${}_{\kappa }$ fan is the same as that of ordinary SLE${}_{\kappa }$.

45 pages, 17 figures

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